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| Description: Elementhood in the domain of cosets. (Contributed by Peter Mazsa, 29-Mar-2019.) | 
| Ref | Expression | 
|---|---|
| eldmcoss | ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dmcoss3 38455 | . . 3 ⊢ dom ≀ 𝑅 = dom ◡𝑅 | |
| 2 | 1 | eleq2i 2832 | . 2 ⊢ (𝐴 ∈ dom ≀ 𝑅 ↔ 𝐴 ∈ dom ◡𝑅) | 
| 3 | eldmcnv 38347 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ◡𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | |
| 4 | 2, 3 | bitrid 283 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ dom ≀ 𝑅 ↔ ∃𝑢 𝑢𝑅𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∃wex 1778 ∈ wcel 2107 class class class wbr 5142 ◡ccnv 5683 dom cdm 5684 ≀ ccoss 38183 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-coss 38413 | 
| This theorem is referenced by: eldmcoss2 38461 eldm1cossres 38462 | 
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